OFFSET
0,4
COMMENTS
Also the number of partitions of n in which each part occurs a triangle number (>=0) times. - Seiichi Manyama, May 11 2018
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from Seiichi Manyama)
FORMULA
Euler transform of A048272. [Vladeta Jovovic, Mar 28 2009]
G.f.: 1/prod(n>=1, P(x^n)^((-1)^(n-1)) ) where P(x) = prod(k>=1, 1-x^k ), see Pari code. [Joerg Arndt, Jun 24 2011]
G.f.: Product_{k>0} (Sum_{m>=0} x^(k*m*(m+1)/2)) = (1+x+x^3+x^6+...)*(1+x^2+x^6+x^12+...)*(1+x^3+x^9+x^18+...)*... . - Seiichi Manyama, May 11 2018
a(n) ~ (log(2))^(3/8) * exp(Pi*sqrt(2*log(2)*n/3)) / (2^(11/8) * 3^(3/8) * Pi^(1/4) * n^(7/8)). - Vaclav Kotesovec, Oct 08 2018
EXAMPLE
From Seiichi Manyama, Mar 11 2018: (Start)
n | Partitions of n in which each part occurs a triangle number (>=0) times.
--+-------------------------------------------------------------------------
1 | 1;
2 | 2;
3 | 3 = 2+1 = 1+1+1;
4 | 4 = 3+1;
5 | 5 = 4+1 = 3+2 = 2+1+1+1;
6 | 6 = 5+1 = 4+2 = 3+2+1 = 3+1+1+1 = 2+2+2 = 1+1+1+1+1+1;
7 | 7 = 6+1 = 5+2 = 4+3 = 4+2+1 = 4+1+1+1 = 2+2+2+1; (End)
MAPLE
with(numtheory):
b:= proc(n) option remember; -add((-1)^d, d=divisors(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..60); # Alois P. Heinz, May 11 2018
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2018 *)
PROG
(PARI) {a(n)=if(n==0, 1, polcoeff(exp(sum(m=1, n, sigma(m)*x^m/(1+x^m+x*O(x^n))/m)), n))}
(PARI) N=99; x='x+O('x^N);
gf=1/prod(n=1, N, eta(x^n)^((-1)^(n-1)));
Vec(gf) /* Joerg Arndt, Jun 24 2011 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 28 2009
STATUS
approved